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Unformatted text preview: probability of S on the n th trial is p n , we can determine whether or not the waiting time variable X = number of trials until the Frst S is proper or not by applying the Sum-Product Lemma below. (The Sum-Product Lemma is a mathematical result from real analysis.) The Sum-Product Lemma: Suppose 0 < p n < 1 for each n . Then p n =1 (1-p n ) = c > if and only if s n =1 p n < application: in the special case (*) noted above, we have P ( X > n ) = P ( FF . . . F ) = (1-p 1 )(1-p 2 ) (1-p n ) . Therefore f ( ) = lim n (1-p 1 )(1-p 2 ) (1-p n ) = P n =1 (1-p n ), and so the Sum-Product Lemma can be applied to determine whether f ( ) > 0 or f ( ) = 0....
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This note was uploaded on 03/17/2012 for the course STAT 333 taught by Professor Chisholm during the Winter '08 term at Waterloo.
- Winter '08